1 MODEL OF RUBBER DEFORMATION Patricia SAAD, Fabrice THOUVEREZ, Jean-Pierre LAINE and Louis JEZEQUEL Laboratoire de Tribologie et Dynamique des Systèmes, Equipe D2S (UMR CNRS 5513) Ecole Centrale de Lyon, BP 163, 69131 Ecully Cedex, France. Phone : 04 72 18 64 64, Fax : 04 72 18 91 44 Email : psaad@mecasola.ec-lyon.fr ABSTRACT A simplified model of rubber bush is presented. Using a Rayleigh Ritz approximation for the displacement field, we compute the static response of a Mooney-Rivlin material. The same procedure can be applied to obtain the frequency response of a viscoelastic bush. In this case, the non linear viscoelasticity, function of the static load and the dynamic displacement level, is modelled using Volterra kernels. The model coefficients (Volterra kernels and Ritz parameters) are identified from experimental data as follows. Complex moduli measures of a rubber bush are carried out for different values of initial static load and dynamic displacement level, then a least squares procedure is applied to fit the experimental data. This new method provides a model with few degrees of freedom that preserves the frequential properties and non linearities of the rubber. NOMENCLATURE [ ] E Green Lagrange strain tensor [] P First Piola Kirchhoff stress tensor [] S Second Piola Kirchhoff stress tensor [ ] M Mass matrix {} U Displacement {} B Reference body force vector W Strain energy Φ External potential energy [ ] F Deformation gradient tensor [ ] [ ][ ] F F C T = Right Cauchy Green tensor [ ] σ Cauchy stress tensor [] s Deviatoric stress tensor [] e Deviatoric strain tensor 1 INTRODUCTION This article proposes a numerical model to compute non linear rubber bush response. The complete model is to be incorporated into full vehicle modelling software. These bushes are rubber made. Elasticity and damping are significant properties of rubber, useful in engineering applications. It is therefore important that the constitutive model accurately capture theses aspects of the mechanical behavior. Hyperelastic strain energies are used to compute quasi-static response of the bush. Theory of linear viscoelasticity is used to model creep, relaxation tests or quasi-static harmonic tests giving hysteresis loops. It can also model frequency domain dynamic behavior of rubber. These models are well adapted to capture the frequency dependance of the dynamic modulus and phase angle. Theory of non linear viscoelasticity must be used to take into account finite deformations, and small harmonic oscillations superposed on a large static predeformation. All these constitutive equations are integrated in finite element models. The main disadvantage of this approach is the complexity of the procedure. The number of dof is too high to be integrated into the framework of a vehicle study. Our work aims at giving a simplified formula of the force as a function of the displacement and its derivatives. The coefficients of the function depend on the geometry of the bush, and on the constitutive equations. 2 GOVERNING EQUATIONS The main specific properties of elastomeric materials are their ability to sustain large deformations without permanent ones and to introduce damping. We need to establish a model for a material with geometrical nonlinearities and non linear constituve laws. Therefore, the large deformation theory will be used [1]. 2.1 3D non linear elastodynamics In the whole document, we use a material description. We consider a structure of initial volume Ω0. The Green Lagrange strain tensor [E] may be expressed in terms of the displacement gradient tensor according to [ ] ( 29 [ ] [ ] NL L T T E E UGradU Grad GradU U Grad E + = + + = 2 1 2 1 (1) The local form of the equation of motion in the reference configuration is [ ] {} {} U B P Div 0 ρ = + (2) 1342